It's this one, I think: https://addons.mozilla.org/firefox/addon/speed-dial-fast/

Their screenshot shows an X close button near the upper right of their "sidebar". If you hover your mouse pointer around in that area, can you find a spot that will close it when clicked? Does pressing the Esc key help?

There's no support address provided on that page, but in the extension, they change your default search provider to Start XXL so perhaps there is an affiliation there that could lead to some help?

I see what you're saying. On mine, I have that little options symbol in the upper right corner. If I click that, the settings section moves over to the left edge of the screen and an X appears, but if I click the X the settings just goes back to where it is now.

Pressing the esc key doesn't do anything.

Hmm...

The part on the left isn't Firefox's sidebar, is it? There's a Sidebars button on the toolbar you can use to show/hide sidebars. Does that show anything interesting?

If that's not it, is your display especially high resolution? Just wondering if there is a reason the speed dial layout might not fill the whole width.

This all most looks like a part of the Firefox Home page that isn't fully covered by the Speed dial page, but just shows a few icons of the Firefox default new tab page.

What happens if you make the Firefox window smaller?

I just checked the sidebars, and it's not one of them. I even turned them all on and off to see if they'd clear this thing out. No luck.

There's no change if I make the Firefox window smaller. everything is still there.

Well, I never found out what the issue was but it appears to have cleared up on its own.